Related papers: Smoothness of Algebraic Supervarieties and Supergr…

Hopf algebras, most generally in a semisimple abelian symmetric monoidal category, are here supposed to be commutative but not to be of finite-type, and their (equivariant) smoothness are discussed. Given a Hopf algebra $H$ in a category…

In this paper, we introduce the solvabilizer and the solvable graph for a Lie superalgebra and establish their basic properties. Then we define a category which links Lie superalgebras to their solvable substructures. Afterwards, we prove…

We prove that every smooth affine variety of dimension $d$ embeds into every simple algebraic group of dimension at least $2d+2$. We do this by establishing the existence of embeddings of smooth affine varieties into the total space of…

We give a definition of the notion of spherical varieties in the world of complex supervarieties with actions of algebraic supergroups. A characterization of affine spherical supervarieties is given which generalizes a characterization in…

We investigate the algebras of invariants and the properties of the quotient morphism by an action of a finite group scheme in terms of stabilizers of points.

We prove that a smooth and connected algebraic group $G$ is affine if and only if any invertible sheaf on any normal $G$-variety is $G$-invariant. For the proof, a key ingredient is the following result: if $G$ is a connected and smooth…

We consider spline functions over simplicial meshes in $\RR^n$. We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of…

We establish two characterizations of an algebraic group scheme $\bigwedge^m GL_n$ over $\mathbb{Z}$. Geometrically, the scheme $\bigwedge^m GL_n$ is a stabilizer of an explicitly given invariant form or, generally, an invariant ideal of…

Let $k$ be an algebraically closed field and let $A$ be a finitely generated $k-$algebra. We show that the scheme of n-dimensional representations of $A$ is smooth when $A$ is hereditary and coherent. We deduce from this the smoothness of…

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using…

We aim at giving a rigorous proof of the state-ments on the smoothness and the dimension of Severi varieties wherethere are gaps in the proofs in some standard literature. The method isa mixture of algebraic and analytic methods.

We apply the supergeometric analogue of Artin's algebraicity criteria to prove algebraicity for four moduli problems in supergeometry: supercurves, super Riemann surfaces, stable supercurves, and stable super Riemann surfaces. The…

We consider an inhomogeneous quantum supergroup which leaves invariant a supersymmetric particle algebra. The quantum sub-supergroups of this inhomogeneous quantum supergroup are investigated.

A fundamental theorem of Barsotti and Chevalley states that every smooth algebraic group over a perfect field is an extension of an abelian variety by a smooth affine algebraic group. In 1956 Rosenlicht gave a short proof of the theorem. In…

We prove that the group $\mathrm{SAut}_{\mathrm{k}}(\mathbb{A}^2)$ is simple as an algebraic group of infinite dimension, over any infinite field $\mathrm{k}$, by proving that any closed normal subgroup is either trivial or the whole group.…

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give…

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite…

We classify deformation quantizations of the symplectic supervarieties that are smooth and admissible. This generalizes the corresponding result of Bezrukavnikov and Kaledin to the super case. We relate the equivalence classes of…

We survey recent results on multiple transitivity of automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the…

We introduce the concept of a homogeneity supermanifold, which is, roughly speaking, a supermanifold equipped with a privileged atlas whose coordinates carry prescribed (real) homogeneity degrees. This structure defines a sheaf of graded…